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Math 130 Worksheet 2: Linear algebra ... but I thought this was a geometry class! For these exercises, I encourage you to work with a partner. Or two partners! 1. Let E ⊂ F ⊂ K all be fields. (You may assume that they are subfields of C). Suppose that {a1 , a2 , ...an } is a basis for F as a vector space over E for K as a vector space over E. ∗ and {b1 , b2 , ...bm } is a basis Show that the set of products of pairs {ai bj | 1 ≤ i ≤ n, 1 ≤ j ≤ m} is a basis for K over F . 2. Translate the statement of problem 1 into a statement about degree of field extensions. ∗ reminder: that means that E is the scalars, and every number x ∈ F can be written as x = e1 a1 + e2 a2 + ... + en an for some numbers ei ∈ E.

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